nLab
phase space

Context

Variational calculus

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

The covariant phase space of a system in physics is the space of all of its solutions to its classical equations of motion, the space of trajectories of the system. Often one considers a parameterization of this by boundary data or choice of a Cauchy surface. This parameterization is what traditionally is just called a “phase space”. The “covariant” in “covariant phase space” is to indicated that it comes without any unnatural choices.

For a system described by Lagrangian mechanics, the covariant phase space comes canonically equipped with a presymplectic structure. A proper phase space or reduced phase space is a quotient space of the covariant phase space on which the presymplectic structure refines to a symplectic structure or Poisson strucure.

Typically these phase spaces are (locally) naturally parameterized by the suitable boundary conditions which uniquely determine the corresponding history of the physical system. Much of the literature on phase spaces deals with parameterizing these boundary conditions.

For instance for a non-relativistic particle propagating on a Riemannian manifold XX with the usual action functional, a trajectory is uniquely fixed by the position xXx \in X and the momentum pT x *Xp \in T^*_x X of the particle at a given time. Correspondingly the space of all solutions and hence the (covariant) phase space of the system may be identified with the cotangent bundle T *XT^* X of XX.

(The term “phase” in “phase space” can be related to the phase of complex numbers in this example, see at phase and phase space in physics.)

However, even reduced phase spaces are not all cotangent bundles, typically not, for instance, if they are obtained by symplectic reduction. This way a finite-dimensional phase space can sometimes describe continuous systems (e.g. in hydrodynamics) whch have infinitely many degrees of freedom; that phase space is however not a cotangent bundle of something in general.

Covariant phase space

There are two main routes to the construction of the covariant phase space,

(S) Via pre-symplectic structures

We describe the canonical presymplectic structure on the covariant phase space of a local action functional. The covariant phase space is defined as the space of critical points of an action functional or, equivalently, the space of solutions of its Euler-Lagrange equations, also known as the shell. The shell is naturally embedded as a subset of the space of all field configurations. Below, terminology and notation are as in the discussion at variational bicomplex.

Let EXE \to X be a smooth fiber bundle over an nn-dimensional spacetime XX and j Ej_\infty E the corresponding jet bundle. The total de Rham differential on j Ej_\infty E decomposes, d=d+δ\mathbf{d} = d + \delta, into the anti-commuting horizontal and vertical differentials, dd and δ\delta. The local action functional

S:Γ(E) S : \Gamma(E) \to \mathbb{R}

is by definition given by a Lagrangian

L:Γ(j E)Ω n(X) L : \Gamma(j_\infty E) \to \Omega^{n}(X)

as

S:ϕ XL(j (ϕ)). S : \phi \mapsto \int_X L(j_\infty(\phi)) \,.

The variation of the action SS can be written in terms of the vertical differential:

δS(ϕ)= X(δL)(j (ϕ)), \delta S (\phi) = \int_X (\delta L)(j_\infty(\phi)) \,,

where the first δ\delta the variational differential (as interpreted in the bicomplex definition) and the second δ\delta is the vertical differential. Thus, for the purposes of variational calculus, we can concentrate purely on the vertical differential of the Lagrangian, which canonically decomposes as follows:

δL=EL(ϕ)δϕdθ(ϕ), \delta L = EL(\phi)\delta\phi - d \theta(\phi) \,,

where EL(ϕ)=0EL(\phi) = 0 is the Euler-Lagrange equation on ϕ\phi, and where EL(ϕ)δϕEL(\phi)\delta\phi is a degree-(n,1)(n,1) and θ\theta is a degree-(n1,1)(n-1,1) element in the variational bicomplex (one variational form degree and, respectively, nn or (n1)(n-1)-spacetime form degrees).

Remark

On a local coordinate patch {x i}\{x^i\} for XX the form θ\theta here is given by

θ(ϕ)=(ι iδLδϕ ,i a)δϕ a. \theta(\phi) = (\iota_{\partial_i} \frac{\delta L}{\delta \phi^a_{,i}} ) \wedge \delta \phi^a \,.

If L=L kin+L potL = L_{kin} + L_{pot} is the sum of a standard kinetic Lagrangian for a free field theory and a potential term that only depends on the fields themselves and not on their derivatives, then this is at the same time the canonical multisymplectic form for the given field bundle. See at multisymplectic geometry the section Examples – Free field theory for more on this relation.

The above definition of ELEL and θ\theta in terms of δL\delta L yields the following identity upon taking another exterior variational derivative of both sides

δ(δL)=0=δ(EL(ϕ))δϕ+dδθ(ϕ), \delta(\delta L) = 0 = \delta(EL(\phi)) \wedge \delta\phi + d \delta \theta(\phi) \,,

where the first term on the right clearly vanishes when pulled back to the shell, EL(ϕ)=0EL(\phi)=0 on XX. The above wedge product between variational forms is inherited from the wedge product on Ω (j E)\Omega^\bullet(j_\infty E). This implies that the presymplectic current, ω(ϕ)=δ(θ(ϕ))\omega(\phi) = \delta(\theta(\phi)), is horizontally closed on shell:

dι *ω(ϕ) =dι *(δθ(ϕ)) =ι *d(δθ(ϕ)) =ι *δ(EL(ϕ))δϕ =0, \begin{aligned} d \iota^* \omega(\phi) &= d \iota^* (\delta \theta(\phi)) \\ &= \iota^* d (\delta \theta(\phi)) \\ &= -\iota^* \delta(EL(\phi)) \wedge \delta\phi \\ &= 0 \, , \end{aligned}

where ι:𝒮 LΓ(E)\iota: \mathcal{S}_L \to \Gamma(E) is the embedding of the solutions of the Euler-Lagrange equations (the shell) into the space of all field configurations. The reason is that ι *δ(EL(ϕ))=0\iota^* \delta(EL(\phi)) = 0 for each xXx\in X, since the functions EL(ϕ)EL(\phi) are all constant (in fact 00) on solutions.

The variational 1-form on the space of field configurations

Θ= X| inθ(ϕ) \Theta = \int_{X|_{in}} \theta(\phi)

given by an integration over a Cauchy surface X| inX|_{in} is the potential for the presymplectic form

Ω=δΘ= X| inδ(θ(ϕ))= X| inω(ϕ) \Omega = \delta \Theta = \int_{X|_{in}} \delta(\theta(\phi)) = \int_{X|_{in}} \omega(\phi)

on the space of field configurations. Since ω(ϕ)\omega(\phi) is horizontally closed on shell, the the pullback of the presymplectic form ι *Ω\iota^* \Omega is independent of the choice of the surface X| inX|_{in} (provided the choice is restricted to a single homology class of surfaces).

Claim

The presymplectic form ι *Ω\iota^* \Omega on the covariant phase space is symplectic iff the linearized Euler-Lagrange equations, EL(ϕ)=0EL(\phi)=0, have a locally well-posed initial value problem on X| inX|_{in}. In particular, in the presence of gauge symmetries, due to the failure of uniqueness of solutions for given initial data on X| inX|_{in}, the form ι *Ω\iota^*\Omega is only presymplectic.

However, the infinitesimal actions of gauge symmetries exhaust the kernel of the ι *Ω\iota^* \Omega and upon performing symplectic reduction, we obtain the space of orbits of solutions under the action of gauge symmetries, which is the physical or reduced phase space.

Notice that the form Ω\Omega, on the field configuration space, does depend on the choice of Cauchy surface. Performing symplectic reduction gives the symplectic space of equivalence classes of solutions of equations of motion modulo gauge transformations, and hence also the reduced phase space. Thus, the end point of the reduction no longer depends on the choice of the Cauchy surface.

Application to the inverse problem of the calculus of variations

We discuss the inverse problem of variational calculus:

given a presymplectic form on the locus of solutions of a system of partial differential equations, when is it the covariant phase space of a local action functional?

(This section follows BridgesHydonLawson.)

We use same notation as the preceding section. Namely dependence on ϕ\phi in local forms really means dependence on finitely many components of the infinite jet j (ϕ)j^\infty(\phi). Also, ι\iota denotes the embedding of the space of solutions in the space of field configurations. Moreover, we presume to work on a sufficiently small neighborhoods in the space of solutions and field configurations that the Poincar'e lemma applies.

Consider a system of partial differential equations P(ϕ)=0P(\phi)=0, together with a local presymplectic form Ω= X| inω(ϕ)\Omega = \int_{X|_{in}} \omega(\phi), where ω(ϕ)\omega(\phi) is a degree-(2,dimX1)(2,dim X-1) element of the variational bicomplex, that is δω=0\delta \omega = 0. Suppose further that presymplectic current density ω(ϕ)\omega(\phi) is horizontally conserved on solutions:

dι *ω(ϕ)=ι *dω(ϕ)=0dω(ϕ)=P(ϕ)λ(ϕ)+δ(P(ϕ))μ(ϕ)=P(ϕ)(λ(ϕ)δμ(ϕ))+δ(P(ϕ)μ(ϕ)), d \iota^* \omega(\phi) = \iota^* d \omega(\phi) = 0 \quad \implies \quad d \omega(\phi) = P(\phi) \lambda(\phi) + \delta(P(\phi)) \wedge \mu(\phi) = P(\phi) (\lambda(\phi) - \delta\mu(\phi)) + \delta (P(\phi)\mu(\phi)) \, ,

where λ\lambda and μ\mu are systems (suitably contracted with the P(ϕ)P(\phi) system) of degree-(2,dimX)(2,dim X) and degree-(1,dimX)(1,dim X) elements of the variational bicomplex. Using the variational closure of ω(ϕ)\omega(\phi), we can conclude that δ(P(ϕ)(λ(ϕ)δμ(ϕ)))=0\delta(P(\phi)(\lambda(\phi)-\delta\mu(\phi))) = 0 and thus, locally, P(ϕ)(λ(ϕ)δ(ϕ))=δ(P(ϕ)λ(ϕ))P(\phi)(\lambda(\phi)-\delta(\phi)) = \delta(P(\phi)\lambda'(\phi)). (…Justify why the representative on the last RHS can be chosen proportional to P(ϕ)P(\phi)…)

On the other hand, the variational closure of ω(ϕ)\omega(\phi) also implies the existence of a degree-(1,dimX1)(1,dim X-1) form θ(ϕ)\theta(\phi), such that δ(θ(ϕ))=ω(ϕ)\delta(\theta(\phi)) = \omega(\phi). The following identity then allows us to (locally) reconstruct a Lagrangian whose Euler-Lagrange equations are satisfied by solutions to P(ϕ)=0P(\phi)=0.

δ(dθ(ϕ)) =d(δθ(ϕ)) =dω(ϕ) =δ[P(ϕ)(λ(ϕ)+μ(ϕ))] dθ(ϕ)δL(ϕ) =P(ϕ)λ(ϕ), \begin{aligned} \delta(d \theta(\phi)) &= - d(\delta\theta(\phi)) \\ &= - d \omega(\phi) \\ &= - \delta[P(\phi)(\lambda'(\phi)+\mu(\phi))] \\ \implies d\theta(\phi) - \delta L(\phi) &= -P(\phi)\lambda''(\phi) \, , \end{aligned}

where λ(ϕ)\lambda''(\phi) is a degree-(1,dimX)(1,dim X) form and L(ϕ)L(\phi) is a Lagrangian degree-(0,dimX)(0,dim X) form. Rearranging the last equality as

δL(ϕ)=P(ϕ)λ(ϕ)+dθ(ϕ), \delta L(\phi) = P(\phi)\lambda''(\phi) + d\theta(\phi) \, ,

we conclude that the Euler-Lagrange equations of L(ϕ)L(\phi) are satisfied on solutions of P(ϕ)=0P(\phi)=0, since EL(ϕ)δϕ=P(ϕ)λ(ϕ)EL(\phi)\delta\phi = P(\phi)\lambda''(\phi).

(P) Via Poisson structures

The covariant phase space can be embedded into the space of field configurations as a subspace of the set of solutions that transversely intersects gauge orbits. This embedding is characterized as the zero locus of the equations of motion and some gauge fixing conditions. The non-degenerate Poisson structure on the algebra of functions on the covariant phase space is given by the Peierls bracket.

The Peierls bracket of two functions AA and BB is the antisymmetrized influence on BB of an infinitesimal perturbation of a gauge-fixed action by function that restricts to AA on the embedding. The algebra of functions on the space of field configurations becomes a Poisson algebra in the following way. Pick a set of functions on the space of field configurations that restrict to a non-degenerate coordinate system on the embedded covariant phase space. These functions, together with the equations of motion and gauge fixing conditions define a Poisson bivector by being declared canonical, such that the kernel of the bivector coincides with the ideal generated by the equations of motion and the gauge fixing conditions. Obviously the Poisson structure thus constructed on the algebra of functions on field configurations is not unique and depends on the above choice of coordinates; the same non-uniqueness may be parametrized instead by a choice of a connection on the space of field configurations. The embedded covariant phase space becomes a leaf of the symplectic foliation of the space of field configurations.

Via the BV-complex

The BV-BRST complex of a local action functional is (the formal dual to) a resolution of the reduced covariant phase space (the quotient of the covariant phase space by symmetries). As discussed in more detail at BV-BRST complex the ghost sector of that complex is a model for the quotient by the symmetries, whereas the antifield/antighost sector is a model for the critical locus of the action functional.

Moreover, by the nature of its construction, the BV-complex is canonically equipped with a graded symplectic form Ω\Omega, whose (-1)-graded Poisson bracket is called the antibracket (essentially the canonical Schouten bracket on graded derivations, see at derived critical locus). This is not the canonical symplectic form Σω\int_\Sigma \omega on the reduced phase space, as discussed above, but it is something like a potential for it.

We want to make the following

Claim

Given a local action functional on a space of fields over a spacetime XX. Let d BVd_{BV} denote the differential of the BV-BRST complex and let dd denote the horizontal de Rham differential on XX. Then

d BV ΣΩ= Xdω. d_{BV} \int_\Sigma \Omega = \int_X d \omega \,.

If XX is a globally hyperbolic spacetime of the form Σ×[0,1]\Sigma \times [0,1] then this is

d BV ΣΩ= Σ outω Σ inω. d_{BV} \int_\Sigma \Omega = \int_{\Sigma_{out}} \omega - \int_{\Sigma_{in}} \omega \,.

We discuss this now in more detail. (The stament then also appears in Cattaneo-Mnev-Reshetikhin 12, equation (9)).

Write

Then the symplectic form density Ω\Omega for the antibracket

ΩΩ loc,BV (n,2)(X×Conf BV) \Omega \in \Omega^{(n,2)}_{loc, BV}(X \times Conf_{BV})

is

Ω= Xd varΦ¯ a(x)d varΦ a(x). \Omega = \int_X d_{var} \bar \Phi_a(x) \wedge d_{var} \Phi^a(x) \,.

Then,

d BV ΣΩ =d BV( Σd varΦ¯ a(x)d varΦ a(x)) =d var(d BV ΣΦ¯ a(x))d varΦ a(x) \begin{aligned} d_{BV} \int_\Sigma \Omega & = d_{BV} \left( \int_\Sigma d_{var} \bar \Phi_a(x) \wedge d_{var} \Phi^a(x) \right) \\ & = - d_{var} (d_{BV} \int_\Sigma \bar \Phi_a(x)) \wedge d_{var} \Phi^a(x) \end{aligned}

We observe that the term in parenthesis is – in the notation at derived critical locus (so we are assuming now the assumptions made there) –

[S^+d BRST,d W(BRST)]=[S^,d W(BRST)]. [\hat S + d_{BRST}, d_{W(BRST)}] = [\hat S, d_{W(BRST)}] \,.

Hence

=d varEL a(x)d varΦ a(x) = Xd varEL, \begin{aligned} \cdots & = - d_{var} EL_a(x) \wedge d_{var} \Phi^a(x) \\ & = - \int_X d_{var} EL \end{aligned} \,,

where ELΩ loc,BV dimX,1(X×Conf BV)EL \in \Omega^{dim X, 1}_{loc, BV}(X \times \mathrm{Conf}_{BV}) is the Euler-Lagrange density as in Zuckerman, p. 267. By equation g) there, this is

= Xdω. \cdots = - \int_X d \omega \,.

Examples

Relativistic particle

The covarint phase space of the relativistic particle on a pseudo-Riemannian manifold XX is the space of geodesics of XX (in the absence of a background gauge field).

Chern-Simons theory

For Chern-Simons theory corresponding to a non-degenerate bilienear invariant polynomial ,\langle -,-\rangle on a Lie algebra 𝔤\mathfrak{g} the

See Chern-Simons theory and ∞-Chern-Simons theory for more details.

In dg-geometry

(…)

Let the ambient context be that of dg-geometry. Let CC be an ordinary smooth manifold, assumed finite dimensional for the moment, and exp(iS):C/\exp(i S) : C \to \mathbb{R}/\mathbb{Z} an ordinary smooth function such that its 0-locus is an sub-manifold.

Then a presentation for the homotopy fiber of exp(iS)\exp(i S) is given by the formal dual of the dg-algebra

( C (X) Γ(TX),ι ()dexp(iS))=[ C (X) 2Γ(TX)ι ()dexp(iS)Γ(TX)ι ()dexp(iS)C (X)] (\wedge^{-\bullet}_{C^\infty(X)}\Gamma(T X), \iota_{(-)} d \exp(i S)) = \left[ \cdots \to \wedge_{C^\infty(X)}^2\Gamma(T X) \stackrel{{\iota_{(-)}d \exp(i S)}}{\to} \Gamma(T X) \stackrel{\iota_{(-)}d \exp(i S)}{\to} C^\infty(X) \right]

concentrated in non-positive degree, which in degree kk has the kkth exterior powers of the tangent vectors of XX and whose differential is given by contracting a tangent vector with the 1-form dexp(iS)d \exp(i S).

This is a Koszul resolution?-type resolution of the 0-locus of exp(iS)\exp(i S). More generally, the homotopy fiber is given by a Koszul-Tate resolution-type complex. This is known as the antifield complex in the BV-BRST formulation of derived phase spaces.

(…)

Reductions of (pre-)symplectic manifolds:

symplectic geometryphysics
presymplectic manifoldcovariant phase space
\downarrow gauge reduction\downarrow quotient by gauge symmetry
symplectic manifoldreduced phase space
\downarrow symplectic reduction\downarrow quotient by global symmetry
symplectic manifoldreduced phase space

References

A a popular introduction to the notion of phase space with some comments on the history of the concept are in

An article reviewing much of the content of the following references is

Covariant phase space

A standard textbook reference is chapter 17.1 of

Reviews of covariant phase space theory include:

A discussion in the context of the variational bicomplex with further pointers to the use in physics is in

A discussion in the language of D-modules, following the book Chiral Algebras and leading up to the derived covariant phase space by BRST-BV formalism is in section 8.3 of

The relation between covariant phase space methods and multisymplectic geometry is discussed in

Original articles on covariant phase space technology

In the following we give a commented list of references following the historical development.

It seems that there have been two independent lines of development, somewhat independent of each other, but also with some crosspollination along the way. One stream (P) was concerned with the construction of a Poisson structure on functions of solutions, while the other (S) was concerned with the construction of a symplectic form on the space of solutions. Obviously, the differences between the two kinds of constructions show up precisely in gauge theory systems.

The earliest papers are those of Peierls and Bergmann-Schiller in the early 50’s. They seem to have been self-motivated. Below, we try to list papers that have established key results or have served as important popularizers together with their main influences from previous works. The constructions of the (S) stream seem to have appeared nearly independently several times over several decades, until Lee-Wald and followups of Ashtekar’s papers became standard references. The (P) stream flowed slowly, but consistently from original idea of Peierls. The papers of Fredenhagen-Dütsch-et-al are a most readable modern formulation.

general relativity as an integral of a conserved form-current over an arbitrary Cauchy surface. Discussion of conversion of symmetries into conserved quantities using this form. Some of the aspects related to the construction of such conserved have apparently been treated in earlier works by Friedman, Sachs and possibly some others.

Reduced phase space

Standard textbooks on classical mechanics include

BV formalism

Discussion via BV-formalism includes